Paper · Dec 27, 2023

Long-time behavior of awesome homogeneous Ricci flows

Roberto Araujo

math.DGarXiv:2312.16517v2Pacific J. Math. 331 (2024) 187-215
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Abstract

We show that the set of awesome homogeneous metrics on non-compact manifolds is Ricci flow invariant. Moreover, if the universal cover of such awesome homogeneous space is not contractible the Ricci flow has finite extinction time, confirming the Dynamical Alekseevskii Conjecture in this case. We also analyze the long-time limits of awesome homogeneous Ricci flows.

Comments: 26 pages. Updated to match the version published in the Pacific Journal of Mathematics

Ricci flowHomogeneous spacesBoundaries & non-compactLong-time existence & convergence

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Finite extinction time of a family of homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Homogeneous spaces, Boundaries & non-compact · Same author Roberto Araujo · Shared terms “finite extinction”, “extinction time”, “Alekseevskii Conjecture”

  2. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo PDF

    Topics Homogeneous spaces, Long-time existence & convergence, Boundaries & non-compact · Same author Roberto Araujo

  3. On the long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi PDF

    Topics Homogeneous spaces, Long-time existence & convergence · Shared terms “time behavior”, “compact manifolds”

  4. The dynamical Alekseevskii conjecture in dimension five

    Maher Billon PDF

    Topics Homogeneous spaces · Shared terms “Alekseevskii Conjecture”, “Dynamical Alekseevskii”

  5. Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

    Weimin Sheng and Jiazhuo YangWider flows PDF

    Topics Homogeneous spaces, Long-time existence & convergence · Shared terms “time behavior”

  6. Long-Time Behavior of the -Ricci Flow on the 2-Torus

    Yifan Guo and Xiaokang Wang PDF

    Topics Long-time existence & convergence · Shared terms “time behavior”, “universal cover”

  7. On the dynamics of positively curved metrics on under the homogeneous Ricci flow

    Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins PDF

    Topics Homogeneous spaces

  8. 3-symmetric spaces, Ricci solitons, and homogeneous structures

    Thomas Murphy and Paul-Andi Nagy PDF

    Topics Homogeneous spaces, Long-time existence & convergence, Boundaries & non-compact

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.